These are lecture notes for AME 60611 Mathematical Methods I, the first of a pair of courses on applied mathematics taught in the Department of Aerospace and Mechanical Engineering of the University of Notre Dame. Most of the students in this course are beginning graduate students in engineering coming from a variety of backgrounds. The course objective is to survey topics in applied mathematics, including multidimensional calculus, ordinary differential equations, perturbation methods, vectors and tensors, linear analysis, linear algebra, and non-linear dynamic systems. In short, the course fully explores linear systems and considers effects of non-linearity, especially those types that can be treated analytically. The companion course, AME 60612, covers complex variables, integral transforms, and partial differential equations.